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Finding coefficients for a power series solution for a differential equation (Trig)

The question and my work are attached. I'm having trouble doing the trig parts since I am trying to get all the sums to have an x^n term so can I build my recurrence relation but I'm stuck at getting the trig stuff to have the x^n term.

Could someone please show me what to do?

Any help would be greatly appreciated!

Thanks in advance!

Re: Finding coefficients for a power series solution for a differential equation (Tri

The solution of the second order DE...

(1)

... is supposed to be analytic in x=0 so that we can write...

(2)

The computation of the derivatives in x=0 is performed as follows. The initial conditions give us and . From (1) we obtain directly...

(3)

Deriving (3) we obtain...

(4)

The computation of the successive derivatives is left to You...

Kind regards